It is desirable for digital communication receivers to monitor estimated bit error rate (BER) while receiving unknown data. A method for accomplishing this is to estimate the ratio of received energy per symbol (Es) to noise power density spectra (No), from which BER may be estimated This ratio (Es/No) can also be used to obtain a measurement of other related operational characteristics, such as the fade depth of a satellite communications link, so that adjustments of the transmitted signal can be made, as necessary, to maintain received signal quality within predefined standards. A popular technique for estimating signal quality based on Es/No estimates is called the "Alpha-Flunk" technique. Its operation and limitations are described below. An improved technique which is the subject of this invention (hereafter called the "Beta-Flunk" technique) is then described in detail.
Typically, as diagrammatically illustrated in FIG. 1, the output of a receiver demodulator 10 is digitized (to some prescribed code resolution) by an analog-digital converter (ADC) 12, which uses a plurality of threshold or quantizing levels, in order to accommodate (soft decision) error correction decoding of received encoded data (via decoder 14). To derive an estimate of Es/No, and thereby bit error rate, the quantized data signal may be coupled to what is known as an "Alpha-flunk" signal processing mechanism executed by signal processor 16, which examines the contents of a selected one of a set of sample bins, each of which extends over a respectively different quantization range. For example, for a quantizer using three quantization thresholds, a first (low level) bin may encompass all symbols that have been quantized with respect to the lowest quantization level. A second (medium level) bin may encompass all symbols quantized with respect to the next lowest or medium level, plus all the symbols quantized in the first bin; namely the second bin is cumulative of the first bin. Also employed is third bin, which encompasses all symbols processed at each of the first, or lowest threshold level, the second, or medium threshold level and a third, or high threshold level, so that like the second bin, the third bin is cumulative (of all three bins). Then, using the count total of a selected one of the (three) bins as a result of quantizing successive received signals, the alpha flunk processor infers the probability of bit error, or bit error rate (BER).
More particularly, for an arbitrary digital communication signal, such as a BPSK signal, and assuming an equal likelihood of the transmission of a `zero ` or a `one`, a probability density function (PDF) of the output of the (analog-digital converter) quantizer may have two weighted normal densities 21, 22, as graphically shown in FIG. 2. The AGC feedback loop of the receiver maintains mean values of the component densities at -1 and +1, as shown, and the standard deviation of both component densities is 1/(2Es/No).sup.1/2. Also depicted in FIG. 2 are uniform stepsize quantizer threshold values Qth occurring at increments of 0.25, for a three bit quantizer.
In general, if the quantizer has a resolution of B bits, then 2.sup.B -1 threshold values I may be denoted by EQU I.sub.(1) &lt;I.sub.(2) &lt;. . . &lt;I.sub.(2.spsb.B-1.sub.-1) &lt;I.sub.(2.spsb.B-1.sub.) =0&lt;I.sub.(2.spsb.B-1.sub.+1) &lt;. . . &lt;I.sub.(2.spsb.B-1.sub.), EQU where EQU I.sub.(2.spsb.B-1.sub.-J) =-I.sub.(2.spsb.B-1.sub.+J),J=1, . . . , 2.sup.B-1 -1.
If the quantizer has uniform stepsize, as is typically the case, then the distance between every pair of adjacent threshold values I is the same. For an exemplary transmission of N unknown symbols (binary digits) and N statistically independent random variables (samples) having the probability density function of FIG. 2, let ##EQU1## then the Alpha-Flunk technique utilizes cumulative sums of these N statistically independent random variables to estimate Es/No (dB). Since the transmitted symbols are unknown, the sign bit may be discarded and the random variables ##EQU2## are utilized to estimate Es/No. The cumulative sums X.sub.(1), X.sub.(1) +X.sub.(2), . . . X.sub.(1) +X.sub.(2) +. . . +X.sub.(2.spsb.B-1.sub.-1) are estimates of the probabilities of the events: ##EQU3## where the 1.multidot.1 notation indicates the "absolute value" operation.
These probabilities, which may be denoted as EQU P.sub.(1), P.sub.(2), . . . , P.sub.(2.spsb.B-1.sub.-1),
respectively, are related to Es/No (dB) through the Q, or error, functions and are plotted, in FIG. 3, versus Es/No (dB) for the quantizer characteristic depicted in FIG. 2. In FIG. 3, the curve leftmost is the usual symbol error rate curve. Thus, there exists a set of relationships between various "pseudo errors" (or "Alpha-Flunks") and Es/No (dB), EQU where Es/No(dB)=g.sub.(i) (P.sub.(i)), i=1, . . . , 2.sup.B-1 -1.
As a result of demodulating N unknown symbols, the random variables or estimates of Es/No (dB) may be expressed as: EQU Es/No(dB).sub.(i) =g.sub.(i) (X.sub.(1) +. . . +X.sub.(i)), i=1, . . . , 2.sup.B-1 -1 (4)
These values are the Alpha-Flunk estimates of Es/No (dB). If N is sufficiently large, Es/No (dB).sub.(i) may be approximated as: EQU Es/No(dB).sub.(i) .apprxeq.D.sub.(i) ((X.sub.(1) +. . . +X.sub.(i))-P.sub.(i)) +Es/No(dB), (5) EQU where: ##EQU4## Equation (6) is represented in FIG. 3 by projection line 25 from the y-axis PDF onto the x-axis through the relationship Es/No (dB)=g.sub.(1) (P.sub.(1)). By a linearizing approximation, the variance of the estimate Es/No (dB).sub.(i) may be expressed as: EQU D.sub.(i).sup.2 P.sub.(i) (1-P.sub.(i))/N. (7)
Each Alpha-Flunk estimate variance depends on the actual value of Es/No (dB) through P.sub.(i) and D.sub.(i). As a result, the best Alpha-Flunk estimate is obtained by choosing the one with the least variance. It should be observed that this is not an exact process, since the actual P.sub.(i) and D.sub.(i) must be estimated through the relationship g.sub.(i) and the observed value of EQU X.sub.(1) +. . . +X.sub.(i).
(Notice that E( EQU X.sub.(1) +. . . +X.sub.(i) =P.sub.(i)). If two estimates have nearly the same variance, the wrong estimate might be chosen. However, the variance estimate error is reasonably insensitive to typical errors in the estimate of P.sub.(i), for the best of the Alpha-Flunk estimates, over a limited range of Es/No. Therefore, a close-to-the-minimum variance (or "best") Alpha-Flunk estimate can be expected through this process, over a limited Es/No range.
A major shortcoming of the above-described "Alpha-Flunk" technique for estimating Es/No (dB) is that it utilizes only a portion of the information available in the probability density characteristic. As a result, the Alpha-Flunk estimator provides accurate results only over a limited range of Es/No values. Using all of the available information would result in more precise estimates of Es/No, as well as an extended dynamic range of the estimator. This latter method, hereafter called Beta-Flunk, is the subject of this invention, as described below.